Logarithmic Functions and Their Graphs Section 3.2 Logarithmic Functions and Their Graphs

Objective By following instructions students will be able to: recognize and evaluate logarithmic functions with base a. How to graph logarithmic functions. How to recognize, evaluate, and graph natural logarithmic functions. Use logarithmic functions to model and solve real-life problems.

Logarithmic Function For a &b are positive, and b is not equal to 0. Logbaseanswer=exponent Baseexponent=answer Read as log base b of a.

Example 1: Evaluate each expression. a) e) b) f) c) d)

Example 2: Use a calculator to evaluate each expression. a) b) c)

U-TRY #1 Evaluate without using a calculator. 1) 2) 3) 4)

Properties of Logarithms logb1=0 because b0=1. 2. logbb=1 because b1=b. 3. logbbx=x and inverse property 4. If logbx=logby, then x=y. One-to-one property

Example 3: Solve each equation for x. a) b)

Example 4: In the same coordinate plane, sketch the graph of each function. a) b)

Example 5: Sketch the graph .

U-TRY #2 Graph in the same coordinate plane. 1) 2)

Transformations of Logarithms function transformation Parent graph Shifts h units to the left Shift h units to the right Shift k units down Shift k units up

Example 6: Describe the transformation of . a) b)

Natural Logarithmic Function The function defined by is called the natural logarithmic function.

Example 7: Use a calculator to evaluate each expression. a) b) c) d)

Properties of Natural Logarithms 1. ln 1=0 because loge1=0. 2. ln e=1 because logee=1. 3. ln ex=x and eln x=x. Inverse properties 4. If ln x = ln y, then x=y. one-to-one property

Example 8: Use the properties of natural logarithms to rewrite each expression. a) b) c) d)

Example 9: Find the domain of each function. a) b) c)

U-TRY #3 Solve the equation for x. 1) 2) 3)

Example 10: Students participating in a psychological experiment attended several lectures on a subject. At the end of the last lecture, and every month for the next year, the students were tested to see how much of the material they remembered. The average scores for the group were given by the human memory model where t is the time in months. What was the average score on the original (t=0) exam? What was the average score at the end of t=2 months? What was the average score at the end of t=6 months?

Revisit Objective Did we… recognize and evaluate logarithmic functions with base a? How to graph logarithmic functions? How to recognize, evaluate, and graph natural logarithmic functions? Use logarithmic functions to model and solve real-life problems?

Homework Pg 236 #s 1-73 EOO